Computational Mathematics for Fluid Dynamics

Spring'2003

MATH-664 (Credit 3)

Instructor: Prabir Daripa
Office: Blocker 629D
Phone Number: 845-1204
Email: daripa@math.tamu.edu
homepage: http://www.math.tamu.edu/$\sim$prabir.daripa

Description: This is an introductory course on computational mathematics for equations of fluid dynamics. The goal in this course is to introduce a variety of numerical methods currently in use for solving problems of fluid dynamics, and also to introduce new ideas that are yet to be explored. Prior to doing this, the instructor will first introduce equations of fluid dynamics in various forms and some essential analytical aspects of these equations and associated partial differential equations which are relevant for their numerical solutions.

Audience: This course can be taken by graduate students from a variety of disciplines including mathematics, almost all engineering departments, and physics.

Prerequisite: Approval of the Instructor. A basic knowledge of linear algebra and vector calculus at the level of Math 311/ Math 601 is essential. Programming skills either in fortran or C or Matlab are also essential. Knowledge of applied partial differential equations at the level of Math 602 is a plus, but is not a prerequisite.

Course Outline:

1. Part I: Basic Equations and Their Basic Properties
   • Computational Fluid Dynamics: Basic philosophy
   • Basic Equations of Fluid Dynamics: Their origin and a discussion of their physical meaning.
   • Partial Differential Equations: Some relevant analytical aspects

2. Part II: Some Numerical Ingredients
   • Basic Aspects of Discretization: Finite difference/volume/element, spectral/element method
   • Grids and grid generation
   • Techniques for domain embedding and adaptivity
   • Domain decomposition methods
   • Multiscale Approach
   • Automatic Mesh Refinement (AMR)
   • Iterative methods, Multigrids, Preconditioners
   • Aspects of serial and parallel algorithms

3. Part III: Computational Techniques
   • Lagrangian (Particle Method), Eulerian and Hybrid Methods for fluid dynamics
   • Various schemes for numerically solving convection-diffusion equations in simple and complicated geometries
   • Various schemes for numerically solving incompressible Navier-Stokes equations in simple and complicated geometries
   • Front/Shock Tracking Methods
   • Shock-Capturing/Level Set Methods

4. Part IV: Student (or group) Specific Computational Projects
   • Free boundary and Interfacial flows
   • Flows invlolving shock-waves, impact of shock waves on solids, shock-shock interaction, shock-turbulence interaction
   • Biological flows/hemodynamics
   • Porous media flows
   • Multiphase flows
   • Flows past many particles
   • Turbulent flows (LES/DNS etc.) and combustion

References

Lectures will be based on various sources including various journal articles and available textbooks on the subject. The first day handout of the class will contain more references which will also be posted on the course homepage which one can access through my homepage given on the first page of this handout.

Grading

Grades of the course will be based on Homeworks, two computational projects and May be one exam (to be decided based on the student input).